"Let $y=y_{1}(x)$ and $y=y_{2}(x)$ be two distinct solutions of the differential equation $\frac{d y}{d x}=x+y$, with $y_{1}(0)=0$ and $y_{2}(0)=1$ respectively. Then, the number of points of intersection of $y=y_{1}(x)$ and $y=y_{2}(x)$ is"

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Accepted Answer

"

${{dy} \over {dx}} = x + y$

Let $x + y = t$

$1 + {{dy} \over {dx}} = {{dt} \over {dx}}$

${{dt} \over {dx}} - 1 = t \Rightarrow \int {{{dt} \over {t + 1}} = \int {dx} }$

$\ln |t + 1| = x + C'$

$|t + 1| = C{e^x}$

$|x + y + 1| = C{e^x}$

For ${y_1}(x),\,{y_1}(0) = 0 \Rightarrow C = 1$

For ${y_2}(x),\,{y_2}(0) = 1 \Rightarrow C = 2$

${y_1}(x)$ is given by $|x + y + 1| = {e^x}$

${y_2}(x)$ is given by $|x + y + 1| = 2{e^x}$

At point of intersection

${e^x} = 2{e^x}$

No solution

So, there is no point of intersection of ${y_1}(x)$ and ${y_2}(x)$.

"

Let $y=y_{1}(x)$ and $y=y_{2}(x)$ be two distinct solutions of the differential equation $\frac{d y}{d x}=x+y$, with $y_{1}(0)=0$ and $y_{2}(0)=1$ respectively. Then, the number of points of intersection of $y=y_{1}(x)$ and $y=y_{2}(x)$ is

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Let $y=y_{1}(x)$ and $y=y_{2}(x)$ be two distinct solutions of the differential equation $\frac{d y}{d x}=x+y$, with $y_{1}(0)=0$ and $y_{2}(0)=1$ respectively. Then, the number of points of intersection of $y=y_{1}(x)$ and $y=y_{2}(x)$ is

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